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How many possible values of m satisfy the inequality |m + 1| – |m – 3| > 4, if m is an integer?
- a)0
- b)1
- c)2
- d)3
- e)4
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
How many possible values of m satisfy the inequality |m + 1| – |...
Understanding the Inequality
We start with the inequality:
|m + 1| – |m – 3| > 4
To solve this, we need to consider the cases based on the critical points where the expressions inside the absolute values change signs, specifically at m = -1 and m = 3.
Case Analysis
1. **Case 1: m < />
- Here, |m + 1| = -(m + 1) and |m - 3| = -(m - 3).
- Substituting into the inequality:
- -m - 1 + m - 3 > 4
- -4 > 4 (which is false)
2. **Case 2: -1 ≤ m < />
- Here, |m + 1| = m + 1 and |m - 3| = -(m - 3).
- Substituting gives:
- m + 1 + m - 3 > 4
- 2m - 2 > 4
- 2m > 6
- m > 3 (not possible in this range)
3. **Case 3: m ≥ 3**
- Here, |m + 1| = m + 1 and |m - 3| = m - 3.
- Substituting gives:
- m + 1 - (m - 3) > 4
- 1 + 3 > 4
- 4 > 4 (which is false)
Conclusion
In all cases, we find that there are no integer values of m that satisfy the inequality |m + 1| – |m – 3| > 4. Thus, the number of integer solutions is:
**0 integer values of m satisfy the inequality.**
The correct answer is option 'A'.
We start with the inequality:
|m + 1| – |m – 3| > 4
To solve this, we need to consider the cases based on the critical points where the expressions inside the absolute values change signs, specifically at m = -1 and m = 3.
Case Analysis
1. **Case 1: m < />
- Here, |m + 1| = -(m + 1) and |m - 3| = -(m - 3).
- Substituting into the inequality:
- -m - 1 + m - 3 > 4
- -4 > 4 (which is false)
2. **Case 2: -1 ≤ m < />
- Here, |m + 1| = m + 1 and |m - 3| = -(m - 3).
- Substituting gives:
- m + 1 + m - 3 > 4
- 2m - 2 > 4
- 2m > 6
- m > 3 (not possible in this range)
3. **Case 3: m ≥ 3**
- Here, |m + 1| = m + 1 and |m - 3| = m - 3.
- Substituting gives:
- m + 1 - (m - 3) > 4
- 1 + 3 > 4
- 4 > 4 (which is false)
Conclusion
In all cases, we find that there are no integer values of m that satisfy the inequality |m + 1| – |m – 3| > 4. Thus, the number of integer solutions is:
**0 integer values of m satisfy the inequality.**
The correct answer is option 'A'.
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Community Answer
How many possible values of m satisfy the inequality |m + 1| – |...
To determine the number of possible values of m that satisfy the inequality |m + 1| - |m - 3| > 4, we can simplify the expression by considering different cases.
Case 1: m < -1
In this case, both m + 1 and m - 3 are negative. Thus, the inequality becomes:
-(m + 1) - (-(m - 3)) > 4
m - 1 + m - 3 > 4
-4 > 4
This inequality is not satisfied, so there are no solutions in this case.
Case 2: -1 ≤ m < 3
In this case, m + 1 is non-negative, and m - 3 is negative. The inequality becomes:
(m + 1) - (-(m - 3)) > 4
m + 1 + m - 3 > 4
2m - 2 > 4
2m > 6
m > 3
Since m must be an integer in this case, there are no values of m that satisfy this inequality.
In this case, both m + 1 and m - 3 are negative. Thus, the inequality becomes:
-(m + 1) - (-(m - 3)) > 4
m - 1 + m - 3 > 4
-4 > 4
This inequality is not satisfied, so there are no solutions in this case.
Case 2: -1 ≤ m < 3
In this case, m + 1 is non-negative, and m - 3 is negative. The inequality becomes:
(m + 1) - (-(m - 3)) > 4
m + 1 + m - 3 > 4
2m - 2 > 4
2m > 6
m > 3
Since m must be an integer in this case, there are no values of m that satisfy this inequality.
Case 3: m ≥ 3
In this case, both m + 1 and m - 3 are non-negative. The inequality becomes:
(m + 1) - (m - 3) > 4
m + 1 - m + 3 > 4
4 > 4
This inequality is not satisfied, so there are no solutions in this case.
In this case, both m + 1 and m - 3 are non-negative. The inequality becomes:
(m + 1) - (m - 3) > 4
m + 1 - m + 3 > 4
4 > 4
This inequality is not satisfied, so there are no solutions in this case.
Based on the analysis of all cases, there are no possible values of m that satisfy the inequality.
Therefore, the correct answer is A: 0.
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